> There is no gender
This is just plainly false. Gender is the social projection of sex, just like a shadow is a projection of an object onto a plane. When you are making claims like "[e]ach of us is born to a sex that is immutable", you are defining and setting up a system of gender.
There is no escape from gender because we live in a society. You don't get to say "there is no shadow". You cannot simply define it away like that.
> But they should not falsely claim the category of a biological sex that is not theirs
There is no falsehood going on, just a different way to set boundaries. One that is better because it leads to greater justice.
> Each of us is born to a sex that is immutable.
This depends entirely on your definition of sex. You have chosen to define sex to mean gametes. I have a different, more expansive definition that also includes sex characteristics. My definition is better than yours because it leads to greater justice in the world.
> The belief that gender exists, or that we have a gender identity that somehow infused the wrong body, is a profoundly religious idea. And by the same token, ungrounded in fact.
What you call religion, I call the massive weight of decades of medical science: https://medicine.yale.edu/lgbtqi/research/gender-affirming-c...
If you do not believe in the legitimacy of transgender identities, it is you who are choosing bigoted religiosity over actual science.